SWhat changed
For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability.
Scientific scope
For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability. This is the explicit sampling/positive-polynomial method, not a second discovery of Weierstrass density. No modern optimal convergence rate, all-function interpolation property, arbitrary interval without rescaling or new probabilistic limit theorem is claimed. An explicit method and proof contribution, not rediscovery of the earlier Weierstrass existence theorem. No optimal modern rate is assigned. Archive bibliographic explanation resolves1912 separate issue versus1913 bound-volume heading; neither creates a second achievement.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This is the explicit sampling/positive-polynomial method, not a second discovery of Weierstrass density. No modern optimal convergence rate, all-function interpolation property, arbitrary interval without rescaling or new probabilistic limit theorem is claimed. An explicit method and proof contribution, not rediscovery of the earlier Weierstrass existence theorem. No optimal modern rate is assigned. Archive bibliographic explanation resolves1912 separate issue versus1913 bound-volume heading; neither creates a second achievement.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.5. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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